# Are Electronic States in a 1D Atomic Chain Eigenstates of the Hamiltonian?

• squareroot
So you can simplify some expressions.In summary, the conversation discusses an atomic chain with one atom in the primitive cell and a lattice constant a, described by the tight binding model. The electronic Hamiltonian is given, and it is assumed that the atomic orbitals are orthonormalized. The conversation then focuses on calculating the eigenstates and eigenvalues of the Hamiltonian in the first Brillouin zone, using a different summation index and the orthogonality of states at different sites.
squareroot

## Homework Statement

1D atomic chain with one atom in the primitive cell and the lattice constant a. The system in described within the tight binding model and contains N-->∞ primitive cells indexed by the integer n. The electronic Hamiltonian is $$H_{0} = \sum_{n} (|n \rangle E_{at} \langle n | -|n+1 \rangle \beta \langle n| - |n \rangle \beta \langle n+1 | )$$

with Eat being the energy on one electron in the state ##|n \rangle ##at site n and## \beta >0 ## represents the energy overlap integral responsable for the interaction between first neighbors. We assume that the atomic orbitals | n \rangle are orthonormalized and neglect the overlap of atomic orbitals on different sites, thus ## \langle n|n' \rangle = \delta_{nn'} ##

First off, show that the electronic states described by :

$$| k \rangle = \frac{1}{\sqrt{N!}}\sum_{n} e^{ikna} |n \rangle$$

are eigenstates of H0 and calculate the corresponding eigenvalues E0(k) in the first Brillouin zone

above

## The Attempt at a Solution

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We start by plugging H0 into the equation $$H_{0}|k \rangle = E_{0} | k \rangle$$ and thus obtaining

$$(\sum_{n} (|n \rangle E_{at} \langle n | -|n+1 \rangle \beta \langle n| - |n \rangle \beta \langle n+1 | ))| k \rangle$$

and now replacing the form of ##|k \rangle"## one gets

$$(\sum_{n} (|n \rangle E_{at} \langle n | -|n+1 \rangle \beta \langle n| - |n \rangle \beta \langle n+1 | ))(\frac{1}{\sqrt{N!}}\sum_{n} e^{ikna} |n \rangle)$$

moving on with

$$H_{0}|k \rangle = \frac{1}{\sqrt(N!)}\sum_{n}|n \rangle E_{at} \langle n |e^{ikna}|n \rangle - |n+1 \rangle \beta \langle n|e^{ikna}|n \rangle - |n \rangle \beta \langle n+1 | e^{ikna}|n \rangle$$

and here is where I get stuck. I don t know how to evaluate ##\langle n |e^{ikna}|n \rangle## and ## |n \rangle \beta \langle n+1 | e^{ikna}|n \rangle ##

From my intuition I think that after solving the LHS of the Schrodinger equation like I started to do above, at one point I should get that the expression above is of form ## Y|k \rangle ## with Y being a number and thus showing that the ket k is a eigenstate of H0.Thank you

You should start by using a different summation index in the Hamiltonian and the state. You used n as the index of sites in the Hamiltonian. Use some other index, say m, when you write |k>. Then use the orthogonality of states at different sites. Also notice that a matrix element like <n| eikna |n> = eikna.

## 1. What is a 1D atomic chain?

A 1D atomic chain is a theoretical model of a linear arrangement of atoms, where the atoms are closely spaced and interact with each other via their electronic orbitals.

## 2. How are "localized states" related to 1D atomic chains?

Localized states refer to electrons that are confined to a specific region of the 1D atomic chain. These states are formed due to the strong interactions between neighboring atoms.

## 3. What are the properties of localized states in a 1D atomic chain?

Localized states have a unique energy level and wave function, and they are highly sensitive to the distance between atoms and the strength of their interactions. They also have a finite lifetime due to their interactions with other electrons and lattice vibrations.

## 4. How do localized states affect the electronic properties of 1D atomic chains?

The presence of localized states can significantly alter the electronic properties of 1D atomic chains. They can create band gaps, affect the electrical conductivity, and lead to the emergence of novel electronic phenomena such as charge density waves and Peierls distortions.

## 5. What are the potential applications of 1D atomic chains and localized states?

1D atomic chains and localized states have the potential to be used in nanoscale electronics and optoelectronics, as well as in the development of quantum computing and sensing devices. They can also provide insights into the behavior of electrons in low-dimensional systems and help us understand the fundamental principles of materials science.

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